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| {{Infobox probability distribution
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| | name = Exponential-Logarithmic distribution (EL)
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| | type = continuous
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| | pdf_image = [[File:Pdf EL.png|300px|Probability density function]]
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| | cdf_image =
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| | notation =
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| | parameters = <math>p\in (0,1)</math><br><math>\beta >0</math>
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| | support = <math>x\in[0,\infty)</math>
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| | pdf = <math>\frac{1}{-\ln p} \times \frac{\beta(1-p) e^{-\beta x}}{1-(1-p) e^{-\beta x}}</math>
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| | cdf = <math>1-\frac{\ln(1-(1-p) e^{-\beta x})}{\ln p}</math>
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| | mean = <math>-\frac{\text{polylog}(2,1-p)}{\beta\ln p}</math>
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| | median = <math>\frac{\ln(1+\sqrt{p})}{\beta}</math>
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| | mode = 0
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| | variance = <math>-\frac{2 \text{polylog}(3,1-p)}{\beta^2\ln p}</math><br> <math>-\frac{ \text{polylog}^2(2,1-p)}{\beta^2\ln^2 p}</math>
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| | skewness =
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| | kurtosis =
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| | entropy =
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| | mgf = <math>-\frac{\beta(1-p)}{\ln p (\beta-t)} \text{hypergeom}_{2,1} </math><br> <math>([1,\frac{\beta-t}{\beta}],[\frac{2\beta-t}{\beta}],1-p)</math>
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| | cf =
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| | pgf =
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| | fisher =
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| }}
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| In [[probability theory]] and [[statistics]], the '''Exponential-Logarithmic (EL)''' distribution is a family of lifetime [[probability distribution|distributions]] with
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| decreasing [[failure rate]], defined on the interval [0, ∞). This distribution is [[Parametric family|parameterized]] by two parameters <math>p\in(0,1)</math> and <math>\beta >0</math>.
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| == Introduction ==
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| The study of lengths of organisms, devices, materials, etc., is of major importance in the [[biological]] and [[engineering]] sciences. In general, the lifetime of a device is expected to exhibit decreasing failure rate (DFR) when its behavior over time is characterized by 'work-hardening' (in engineering terms) or 'immunity' (in biological terms).
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| The exponential-logarithmic model, together with its various properties, are studied by Tahmasbi and Rezaei (2008)<ref name="tahmasbi2008">Tahmasbi, R., Rezaei, S., (2008), "A two-parameter lifetime distribution with decreasing failure rate", ''Computational Statistics and Data Analysis'', 52 (8), 3889-3901. {{doi|10.1016/j.csda.2007.12.002}}</ref>
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| This model is obtained under the concept of population heterogeneity (through the process of
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| compounding).
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| == Properties of the distribution ==
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| === Distribution ===
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| The [[probability density function]] (pdf) of the EL distribution is given by Tahmasbi and Rezaei (2008)<ref name="tahmasbi2008"/>
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| :<math> f(x; p, \beta) := \left( \frac{1}{-\ln p}\right) \frac{\beta(1-p)e^{-\beta x}}{1-(1-p)e^{-\beta x}} </math>
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| where <math>p\in (0,1)</math> and <math>\beta >0</math>. This function is strictly decreasing in <math>x</math> and tends to zero as <math>x\rightarrow \infty</math>. The EL distribution has its [[Mode (statistics)|modal value]] of the density at x=0, given by
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| :<math>\frac{\beta (1-p)}{-p \ln p}</math>
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| The EL reduces to the [[exponential distribution]] with rate parameter <math>\beta</math>, as <math>p\rightarrow 1</math>.
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| The [[cumulative distribution function]] is given by
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| :<math>F(x;p,\beta)=1-\frac{\ln(1-(1-p) e^{-\beta x})}{\ln p},</math>
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| and hence, the [[median]] is given by
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| :<math>x_\text{median}=\frac{\ln(1+\sqrt{p})}{\beta}</math>.
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| === Moments ===
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| The [[moment generating function]] of <math>X</math> can be determined from the pdf by direct integration and is given by
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| : <math>M_X(t) = E(e^{tX}) = -\frac{\beta(1-p)}{\ln p (\beta-t)} F_{2,1}\left(\left[1,\frac{\beta-t}{\beta}\right],\left[\frac{2\beta-t}{\beta}\right],1-p\right),</math>
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| where <math>F_{2,1} </math> is a [[hypergeometric function]]. This function is also known as ''Barnes's extended hypergeometric function''. The definition of <math>F_{N,D}({n,d},z)</math> is
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| : <math>F_{N,D}(n,d,z):=\sum_{k=0}^\infty \frac{ z^k \prod_{i=1}^p\Gamma(n_i+k)\Gamma^{-1}(n_i)}{\Gamma(k+1)\prod_{i=1}^q\Gamma(d_i+k)\Gamma^{-1}(d_i)}</math>
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| where <math>n=[n_1, n_2,\dots , n_N]</math> and <math>{d}=[d_1, d_2, \dots , d_D]</math>.
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| The moments of <math>X</math> can be derived from <math>M_X(t)</math>. For
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| <math>r\in\mathbb{N}</math>, the raw moments are given by
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| :<math>E(X^r;p,\beta)=-r!\frac{\operatorname{Li}_{r+1}(1-p) }{\beta^r\ln p},</math>
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| where <math>\operatorname{Li}_a(z)</math> is the [[polylogarithm]] function which is defined as
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| follows:<ref>Lewin, L. (1981) ''Polylogarithms and Associated Functions'', North
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| Holland, Amsterdam.</ref>
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| :<math>\operatorname{Li}_a(z) =\sum_{k=1}^{\infty}\frac{z^k}{k^a}.</math>
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| Hence the [[mean]] and [[variance]] of the EL distribution
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| are given, respectively, by
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| :<math>E(X)=-\frac{\operatorname{Li}_2(1-p)}{\beta\ln p},</math>
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| :<math>\operatorname{Var}(X)=-\frac{2 \operatorname{Li}_3(1-p)}{\beta^2\ln p}-\left(\frac{ \operatorname{Li}_2(1-p)}{\beta\ln p}\right)^2.</math>
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| === The survival, hazard and mean residual life functions ===
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| [[File:Hazard EL.png|thumb|300px|Hazard function]]
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| The [[survival function]] (also known as the reliability
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| function) and [[hazard function]] (also known as the failure rate
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| function) of the EL distribution are given, respectively, by
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| : <math>s(x)=\frac{\ln(1-(1-p)e^{-\beta x})}{\ln p},</math>
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| : <math>h(x)=\frac{-\beta(1-p)e^{-\beta x}}{(1-(1-p)e^{-\beta x})\ln(1-(1-p)e^{-\beta x})}.</math>
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| The mean residual lifetime of the EL distribution is given by
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| : <math>m(x_0;p,\beta)=E(X-x_0|X\geq x_0;\beta,p)=-\frac{\operatorname{Li}_2(1-(1-p)e^{-\beta x_0})}{\beta \ln(1-(1-p)e^{-\beta x_0})}</math>
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| where <math>\operatorname{Li}_2</math> is the [[dilogarithm]] function
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| === Random number generation ===
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| Let ''U'' be a [[random variate]] from the standard [[Uniform distribution (continuous)|uniform distribution]].
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| Then the following transformation of ''U'' has the EL distribution with
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| parameters ''p'' and ''β'':
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| : <math> X = \frac{1}{\beta}\ln \left(\frac{1-p}{1-p^U}\right).</math>
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| == Estimation of the parameters ==
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| To estimate the parameters, the [[Expectation-maximization algorithm|EM algorithm]] is used. This method is discussed by Tahmasbi and Rezaei (2008).<ref name="tahmasbi2008"/> The EM iteration is given by
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| : <math>\beta^{(h+1)} = n \left( \sum_{i=1}^n\frac{x_i}{1-(1-p^{(h)})e^{-\beta^{(h)}x_i}} \right)^{-1},</math>
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| : <math>p^{(h+1)}=\frac{-n(1-p^{(h+1)})} { \ln( p^{(h+1)}) \sum_{i=1}^n
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| \{1-(1-p^{(h)})e^{-\beta^{(h)} x_i}\}^{-1}}.</math>
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| ==Related distributions==
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| The EL distribution has been generalized to form the Weibull-logarithmic distribution.<ref>Ciumara1,Roxana; Preda2, Vasile (2009) [http://www.vgtu.lt/leidiniai/leidykla/ASMDA_2009/PDF/16_sec_081_Ciumara_The_Weibull.pdf "The Weibull-logarithmic distribution in lifetime analysis and its properties"]. In: L. Sakalauskas, C. Skiadas and
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| E. K. Zavadskas (Eds.) [http://www.vgtu.lt/leidiniai/leidykla/ASMDA_2009/ ''Applied Stochastic Models and Data Analysis''], The XIII International Conference, Selected papers. Vilnius, 2009 ISBN 978-9955-28-463-5</ref>
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| If ''X'' is defined to be the [[random variable]] which is the minimum of ''N'' independent realisations from an [[exponential distribution]] with rate paramerter ''β'', and if ''N'' is a realisation from a [[logarithmic distribution]] (where the parameter ''p'' in the usual parameterisation is replaced by {{nowrap|1=(1 − ''p'')}}), then ''X'' has the exponential-logarithmic distribution in the parameterisation used above.
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| ==References==
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| {{Reflist}}
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| {{ProbDistributions|continuous-semi-infinite}}
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| [[Category:Continuous distributions]]
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| [[Category:Survival analysis]]
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| [[Category:Probability distributions]]
| |
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